Moment of Inertia Calculator
Calculate the second moment of area for common cross-section shapes.
Results
The second moment of area (moment of inertia) determines a beam's resistance to bending. Higher values mean less deflection and lower stress under load. Shape matters enormously - an I-beam can have 10x the moment of inertia of a solid rectangle with the same cross-sectional area. This calculator handles rectangles, solid circles and hollow tubes.
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Frequently asked questions
I = bh³/12 = 100 × 50³/12 = 1,041,667 mm⁴ or 104.17 cm⁴. The height matters much more than width because it is cubed.
I depends on h³, so doubling the height increases I by 8×, while doubling the width only doubles it. A 100 × 100 mm beam has 4× the I of a 200 × 50 mm beam.
Subtract the inner circle's I from the outer: I = π(D⁴-d⁴)/64. A tube with D=100 mm, d=90 mm has I = π(100⁴-90⁴)/64 = 1,688,000 mm⁴. Hollow sections are very efficient.
Section modulus S = I/c where c is the distance to the outer fiber. For a rectangle, S = bh²/6. Bending stress σ = M/S, so higher S means lower stress.
I-beams concentrate material far from the neutral axis where it contributes most to I. A W200×46 I-beam has I = 45,800 cm⁴ but only weighs 46 kg/m, while a solid rectangle with the same I would weigh about 120 kg/m.